Summary: DIRECTIONAL ROUTING VIA GENERALIZED st-NUMBERINGS
FRED S. ANNEXSTEIN AND KENNETH A. BERMAN
SIAM J. DISCRETE MATH. c 2000 Society for Industrial and Applied Mathematics
Vol. 13, No. 2, pp. 268­279
Abstract. We present a mathematical model for network routing based on generating paths
in a consistent direction. Our development is based on an algebraic and geometric framework for
defining a directional coordinate system for real vector spaces. Our model, which generalizes graph
st-numberings, is based on mapping the nodes of a network to points in multidimensional space and
ensures that the paths generated in different directions from the same source are node-disjoint. Such
directional embeddings encode the global disjoint path structure with very simple local information.
We prove that all 3-connected graphs have 3-directional embeddings in the plane so that each node
outside a set of extreme nodes has a neighbor in each of the three directional regions defined in
the plane. We conjecture that the result generalizes to k-connected graphs. We also show that a
directed acyclic graph (dag) that is k-connected to a set of sinks has a k-directional embedding in
(k - 1)-space with the sink set as the extreme nodes.
Key words. graph connectivity, network routing, st-numbering, matchings
AMS subject classifications. 68R10, 05C40, 68R10
PII. S0895480198333290
1. Introduction. A fundamental problem in network routing is the generation
of communication paths from a set of source nodes Y to a set of sink nodes X. Routing